Effect of Stress Level on Response of Model Monopile to Cyclic Lateral Loading in Sand

Richards, I. A.; Bransby, M. F.; Byrne, B. W.; Gaudin, C.; Houlsby, G. T.

doi:10.1061/(ASCE)GT.1943-5606.0002447

Open access

Technical Papers

Jan 4, 2021

Effect of Stress Level on Response of Model Monopile to Cyclic Lateral Loading in Sand

Authors: I. A. Richards [email protected], M. F. Bransby https://orcid.org/0000-0001-8444-5995, B. W. Byrne, C. Gaudin, and G. T. HoulsbyAuthor Affiliations

Publication: Journal of Geotechnical and Geoenvironmental Engineering

Volume 147, Issue 3

https://doi.org/10.1061/(ASCE)GT.1943-5606.0002447

PDF

Abstract

Monopile foundations supporting offshore wind turbines are exposed to cyclic lateral loading, which can cause accumulated pile displacement or rotation and evolution of the dynamic response. To inform the development of improved design methods, the monopile’s response to cyclic lateral loading has been explored through small-scale physical modeling at

1 g

and in the centrifuge, as well as at large-scale in the field. There are advantages and disadvantages to each physical modeling technique, and the response may be most efficiently explored through a combination of modeling techniques. However, stress levels vary significantly between these techniques, and only centrifuge testing can simulate full-scale stress levels. This paper explores the effect of stress level on the response of a monopile foundation in dry sand to monotonic, unidirectional cyclic and multidirectional cyclic lateral loading with small-scale tests at

1 g

and in the centrifuge at

9 g

and

80 g

. With an identical setup at each

g - level

, stress-level effects were isolated. Qualitatively, the responses are similar across the stress levels, but some important quantitative differences are revealed. In particular, the rate of accumulation of pile displacement and the rate of change of secant stiffness under cyclic loading are found to reduce with increasing stress level. The results highlight the need to simulate full-scale stress levels to thoroughly understand foundation behavior, but also demonstrate the qualitative insight that can be gained through

1 g

physical modeling. The data and trends presented in this paper provide input for the modeling of monopile responses at different stress levels.

Introduction

Monopiles are the principal foundation solution for offshore wind turbines (OWTs) in shallow to medium water depths and currently support 87% of OWTs in Europe (Wind Europe 2018). The response of these large-diameter hollow steel piles has been the subject of much recent research because their geometry and load regime differ from piles traditionally used for offshore oil and gas and onshore applications. Monopiles are subjected to long-term multidirectional cyclic lateral loading caused by a combination of wind, waves, and current loads, which has been shown in model tests to cause accumulated displacement or rotation of the foundation (e.g., Li et al. 2010; Truong et al. 2019) and evolution of foundation stiffness (Leblanc et al. 2010) and energy dissipation (Abadie et al. 2019). Accumulated rotation of the foundation is of particular concern because turbine manufacturers specify strict rotation limits, and changes to foundation stiffness and energy dissipation impact the dynamic response of the foundation and can enhance dynamic amplification of loads and increase fatigue damage of the OWT structure (Bhattacharya 2014).

Given that monopiles may be up to 8 m in diameter (Sørensen et al. 2017), with the latest designs being even larger, it is neither practical nor economic to test monopiles at full scale. However, physical modeling—at

1 g

in the centrifuge and at large (but reduced) scale in the field—has been used extensively to explore the response of a monopile. Because cyclic tests often involve many thousands of cycles and need to be performed under load control, many cyclic campaigns have been performed at

1 g

where full-scale stress levels are not simulated (e.g., Leblanc et al. 2010; Nicolai and Ibsen 2014; Arshad and O’Kelly 2017). Some studies have explored a monopile’s cyclic response in the centrifuge, simulating full-scale stress levels, although the number of cycles is limited (e.g., Li et al. 2010; Klinkvort and Hededal 2013; Truong et al. 2019). Large-scale field testing has also been performed to understand a monopile’s response (principally under monotonic loading) (Byrne et al. 2017). These tests provide valuable insight with stress level, grain size, and installation method closer to that at full scale; however, large-scale tests are very costly and there is intrinsic residual uncertainty about the soil conditions at each pile location.

Understanding a monopile’s response to realistic multidirectional cyclic lateral loading is necessary for the development of new design methods for the next generation of monopiles. Given the complexity of the problem, it may be most efficiently explored through a combination of

1 g

tests, centrifuge tests, and large-scale field tests. However, stress levels vary significantly among these test types, and only centrifuge tests are able to simulate the full-scale stress regime. Understanding the impact of stress level on a monopile’s cyclic response is therefore an important step in bringing together observations from various test types. Other parameters such as grain size and installation method will also vary among these test types, but this work focuses exclusively on stress level.

The fundamental behavior of sand depends on the stress level. For example, dilatancy and peak friction angle

ϕ_{p}^{'}

increase approximately logarithmically with reducing stress level (Bolton 1986), whereas the maximum shear modulus

G_{MAX}

increases as a power function of stress level, with the exponent typically 0.5 [first reported for angular grains by Hardin (1965), with a more recent review by Oztoprak and Bolton (2013)]. This stress-level-dependent behavior is reflected in the global foundation response, as observed by Ovesen (1975) for footings and Klinkvort (2012) for monotonically laterally loaded monopiles.

Few studies have explored the impact of stress level on a monopile’s response to cyclic lateral loading despite the prevalence of

1 g

testing in this area. Rudolph et al. (2014a, b) reported a decrease in accumulated displacement for centrifuge tests compared with

1 g

tests, and Nicolai et al. (2017) compared postcyclic behavior between

1 g

and centrifuge tests. However, the setup and sand type differed between

1 g

and centrifuge tests in these studies, and so stress-level effects were not completely isolated.

This paper extends the work of Richards et al. (2019) and specifically addresses the important issue of stress level, presenting the response of a monopile in dry sand to monotonic, unidirectional cyclic and multidirectional cyclic loading at three

g

-levels. Tests were performed in the

5 - m - radius

beam centrifuge at the University of Western Australia (UWA) at

9 g

and

80 g

, with corresponding

1 g

tests performed on the laboratory floor. By varying the

g

-level, effective unit weight and stress level were varied; with an identical setup at each

g

-level, stress-level effects were isolated from other phenomena.

Experimental Setup

Sample Preparation

UWA superfine (SF) silica sand obtained from Sibelco, Perth, Australia was used for these tests, with properties as summarized in Table 1. Samples were prepared by air pluviation into a square strongbox with base dimensions of

996 \times 996 mm

to a sample height of 376 mm. Three dense samples were prepared with an average unit weight

γ_{a v}^{'} = 17.00 \pm 0.20 kN / m^{3}

(average relative density

D_{R, a v} = 87.4 % \pm 5 %

). The samples were prepared dry to ensure a fully drained response during pile loading.

Table 1. Properties of UWA SF sand

Property	Notation	Value	Unit
Specific gravity	$G_{s}$	2.67	—
Particle size	$d_{10}, d_{50}, d_{60}$	0.12, 0.18, 0.19	mm
Minimum effective unit weight	$γ_{MIN}^{'}$	14.69	$kN / m^{3}$
Maximum effective unit weight	$γ_{MAX}^{'}$	17.40	$kN / m^{3}$
Maximum void ratio	$e_{MAX}$	0.78	—
Minimum void ratio	$e_{MIN}$	0.51	—
Critical state friction angle (triaxial testing)	$ϕ_{CS}^{'}$	31.9	Degrees

Source: Data from Chow et al. (2018).

The piles were installed wished-in-place during the sand raining process. Although it is acknowledged that this installation method is not representative of full-scale monopile installation and that the pile installation method can affect the lateral stiffness and bearing capacity (e.g., Klinkvort 2012; Fan et al. 2019), this method avoids the introduction of complex stress fields and local density changes through in-flight (or

1 g

) installation, which may scale differently from the postinstallation response investigated here.

Sand was first rained to the depth of the pile tip before hanging the piles in position and raining around them. The sand surface was then vacuumed to achieve an average pile embedment

L_{a v} = 167 \pm 3 mm

. Nine piles were installed per strongbox, with a minimum center-to-center spacing of

7.4 D

. This layout was chosen to avoid significant boundary effects while maximizing the number of tests per strongbox sample; the center-to-center spacing is in line with previous studies (e.g., Leblanc et al. 2010; Abadie 2015; Albiker et al. 2017).

Cone penetration tests (CPTs) were performed using a

7 - mm - diameter

cone to characterize each soil sample. For Sample S1, used for

1 g

testing, two CPTs were performed at each stage: (a) before testing, and (b) after testing. For Samples S2 and S3, used for testing at

9 g

and

80 g

, respectively, one CPT was performed at each stage: (a) at

1 g

before spinning, (b) at

n g

before testing, (c) at

n g

after testing, and (d) at

1 g

after spinning down. Fig. 1 presents the CPT profiles measured at each

g

-level. At

1 g

, the S1 CPTs and prespinning S2 and S3 CPTs show good consistency, in line with the small variation in global unit weight measured across the samples. The consistency of the pretesting and posttesting S2 and S3 CPTs also gives confidence in the homogeneity of the sample and insensitivity to repeated spin cycles associated with each monopile test. The postspinning S2 and S3 CPTs show increased CPT resistance, which may be attributed to overconsolidation effects given that these samples previously experienced higher stress levels (Gui et al. 1998; Roy et al. 2019). However, no monopile tests were performed on these overconsolidated samples.

Model Pile, Loading, and Instrumentation

Tests were performed on sandblasted mild steel piles with properties summarized in Table 2. The small

L / D

ratio is representative of modern full-scale monopiles (Sørensen et al. 2017; Schroeder et al. 2015), whereas the large

h / D

ratio is expected to be representative of operational loading conditions (Richards 2019) and ensures a rotation-dominated response. The surface is classified as smooth on the basis of the normalized roughness

R_{n}

(e.g., Hu and Pu 2004), in contrast to typically rough full-scale monopiles. However, roughness is likely to have only a small impact on the pile’s lateral response (Klinkvort 2012). The pile has a closed end, which is appropriate given the wished-in-place installation method. Although full-scale monopiles are open-ended, the contribution of base shear and base moment is typically small (Burd et al. 2020). The pile wall thickness is selected to ensure rigid behavior at all three stress levels; such rigid pile behavior is expected to be representative for modern monopiles (e.g., Abadie 2015).

Table 2. Model pile properties

Property	Notation	Value	Unit
Outside diameter	$D$	42	mm
Target embedded length	$L$	168	mm
Length to diameter ratio	$L / D$	4	—
Wall thickness	$t$	3.2	mm
Eccentricity	$h$	424	mm
Eccentricity to diameter ratio	$h / D$	10	—
Mean surface roughness	$R_{a}$	3	$μ m$
Normalized roughness	$R_{n} = R_{a} / d_{50}$	0.016	—

Fig. 2 shows the loading and instrumentation apparatus, which was modified from that designed by Herduin (2019) for multidirectional loading of anchors. Actuators sit perpendicular to Platform A and apply load to the pile via cables, which travel around pulleys on Platform C and attach to in-line load cells at the base of the pile stickup. Pile displacement was measured with six string potentiometers. Three were positioned 253 mm above the load application point (on Platform B), and three were positioned 30 mm above the load application point (on Platform C). Each triplet of string potentiometers was arranged in a 120° star to attempt to eliminate the net load applied to the pile by these sensors because each applies a tensile load of

\sim 1 N

in the direction of action. The pile stick-up allowed straightforward connection of the string potentiometers and load lines to the pile. The stick-up consists of an insert that is fixed inside the pile head and a threaded rod to which the potentiometers and load lines were attached.

Fig. 2. Schematic (not to scale) and photograph of loading and pile measurement apparatus.

The setup shown in Fig. 2 is for multidirectional loading, with three actuators and associated load lines positioned 120° apart. For unidirectional loading, only two actuators were used, positioned 180° apart, to simplify the setup. All cyclic tests were performed under load control with sinusoidal waveforms, whereas initial monotonic tests were performed under displacement control. Under load control, all load lines were kept in tension, with one or two lines typically holding constant load and one line applying cyclic loading.

To resolve pile position, vertical displacements were neglected and the planar position of the pile at the level of each triplet of string potentiometers was calculated. Under unidirectional loading, pile displacement was obtained directly from the string potentiometer aligned with the loading direction. Under multidirectional loading, measurements from all three potentiometers were used, and to account for the measurement redundancy, the error between actual and measured length was assumed to be equal for all three string potentiometers. Net loads were obtained directly under unidirectional loading, whereas under multidirectional loading, they were resolved to account for the current pile position and load line angle.

Methodology

In this study, the stress level was controlled by varying the model

g

-level using the centrifuge. With the same setup at each

g

-level, stress-level effects were isolated from other variables. The

g - levels

were chosen to vary logarithmically in line with the expected logarithmic variation of dilatancy with stress level.

The vertical effective stress at 70% pile embedment was used as a reference effective stress level (

σ_{REF}^{'} = 0.7 L γ^{'}

) and as a proxy for the mean effective stress

p^{'}

. For the model and sand sample (

γ_{M}^{'} = 17 kN / m^{3}

) used, Table 3 presents the reference effective stress level and normalized effective stress level (

σ_{REF}^{'} / p_{a}

) corresponding to each

g - level

. Table 3 also presents the prototype monopile diameter

D_{P}

simulated at each

g - level

, assuming a dense saturated sand at prototype-scale (

γ_{P}^{'} = 10 kN / m^{3}

). Fig. 3 shows how the model tests translate to prototype scale. Although modern monopiles are even larger (

6 m < D < 10 m

) than that simulated by the

80 g

tests, the range of stress levels is sufficiently large to observe relevant stress-level effects.

Table 3. Corresponding

g

-levels, normalized stress levels, and prototype pile diameters

$g - level$ , $n$	$σ_{REF}^{'}$ (kPa)	$σ_{REF}^{'} / p_{a}$	$D_{P}$ (m)
1	2	0.02	0.071
9	18	0.18	0.64
80	162	1.60	5.7

Fig. 3. Location of model tests at $n g$ and equivalent prototypes at $1 g$ relative to current full-scale monopiles.

Modeling of models campaigns (e.g., Ovesen 1975; Dewoolkar et al. 1999; Klinkvort et al. 2013) are more common than stress-level investigations and involve tests at various

g

-levels and model sizes to simulate the same prototype response to explore scale effects or verify centrifuge modeling techniques. An example modeling of models campaign performed by Klinkvort et al. (2013) is shown in Fig. 3 for context.

Monotonic Response

Fig. 4 presents the response for the monotonic tests alongside the responses for the initial loading portion of the unidirectional cyclic tests. Fig. 4 highlights the significant variation in response as the stress level is varied, as well as the consistency of behavior at each stress level.

Data are presented in terms of applied horizontal load

H

and pile displacement at the load application point

u

(both at model scale). The imperfectly rigid connection between the pile and pile stickup introduces some error into the resolved pile rotation and prevents presentation of the response in terms of applied moment

M

and pile rotation

θ

. However, conclusions on stress-level effects are not expected to be affected by the choice of either work-conjugate pair.

Normalization Approaches

Casting the monopile’s response in dimensionless form facilitates comparison of tests conducted across this large range of stress levels and can help provide insight into stress-level effects. Three different normalization approaches are considered following (1) Klinkvort et al. (2013), (2) Klinkvort (2012), and (3) Leblanc et al. (2010), as summarized in Table 4.

Table 4. Comparison of normalization approaches

Parameter	Normalization 1(Klinkvort et al. 2013)	Normalization 2(Klinkvort 2012)	Normalization 3 (Leblanc et al. 2010; Abadie 2015)
Load	$\frac{H}{D^{3} γ^{'}}$	$\frac{H}{K_{p} D^{3} γ^{'}}$	$\frac{H}{L^{2} D γ^{'}}$
Load	$\frac{H}{D^{3} γ^{'}}$	$K_{p} = \tan^{2} (45 + ϕ_{p}^{'} / 2)$	$\frac{H}{L^{2} D γ^{'}}$
Displacement	$u / D$	$u / D$	$\frac{u}{D} {(\frac{p_{a}}{L γ^{'}})}^{η - 1}$
Displacement	$u / D$	$u / D$	$η = 0.5$

Fig. 5(a) shows the result of applying Normalization 1, which accounts for variation of

γ^{'}

(but not for changes in stiffness or angle of friction) to the monotonic responses. This normalization therefore highlights stress-level effects: the decrease in normalized initial stiffness and inferred normalized capacity with increasing stress-level is indicative of stress dependent stiffness and dilatant behavior.

Fig. 5. Normalization of monotonic responses (lower plots highlighting initial response): (a) method presented by Klinkvort et al. (2013); and (b) method presented by Leblanc et al. (2010).

When exploring stress level effects, Klinkvort (2012) proposed a further approach (Normalization 2), which introduces Rankine’s passive earth pressure coefficient

K_{p}

to attempt to account for stress dependent changes in peak angle of friction

ϕ_{p}^{'}

. Klinkvort (2012) used this normalization to successfully collapse the response of a centrifuge model monopile at stress levels from

\sim 70

to

\sim 350 kPa

, with

ϕ_{p}^{'}

values obtained from complementary triaxial testing. Complementary triaxial tests are not available for this study, and moreover, obtaining definitive

ϕ_{p}^{'}

values at stress levels corresponding to the

1 g

tests (2 kPa) is likely to be very challenging. A number of relationships exist that allow

ϕ_{p}^{'}

to be estimated for a given stress level and relative density (Bolton 1986, 1987; Chakraborty and Salgado 2010), but there is disagreement at low stress levels.

The relationship proposed by Bolton (1987), which limits dilatancy at low stress levels, is employed in this work. This relationship is supported by the work of e.g., Tatsuoka et al. (1986) and the recent investigation into the behavior of Leighton Buzzard sand by White (2020), which found that although the shear strength properties (e.g.,

ϕ_{p}^{'}

) tend to increase with decreasing effective confining stress, this effect becomes increasingly minor for confining stresses less than approximately

50 kN / m^{2}

. However, other studies (Ponce and Bell 1971; Quinteros et al. 2017) have presented conflicting results. Employing the relationship of Bolton (1987) to estimate

ϕ_{p}^{'}

for Normalization 2,

K_{p}

changes negligibly over the stress range of concern here, and the plot resembles Fig. 5(a).

In contrast to the approach of Klinkvort (2012), Leblanc et al. (2010) incorporated stress dependent stiffness into Normalization 3, but did not consider dilatancy (i.e., stress-level-dependent changes in angle of friction). Stress dependent stiffness is incorporated by assuming

G \propto p^{' η}

. The exponent is chosen as

η = 0.5

, which is generally accepted at small strains (

γ < 0.01 %

) i.e.,

G_{MAX} \propto p^{' 0.5}

(e.g., Hardin 1965). At larger strains,

η \to 1

(Oztoprak and Bolton 2013). Fig. 5(b) shows the results of applying Normalization 3. The normalization successfully collapses the results toward a single curve, particularly at the small displacements that are relevant to the cyclic tests and which are expected to be controlled by small strain stiffness.

Together, the normalizations indicate the presence of stress-level-dependent stiffness and possibly stress-level-dependent dilatant behavior, although it is not possible to decouple these phenomena in this study. The normalization approach of Leblanc et al. (2010) is adopted here for comparison of responses across stress levels.

Maximum Stiffness

The monopile’s maximum tangential stiffness

k_{MAX}

is expected to vary with stress level in the same way as the soil’s maximum shear modulus

G_{M A X}

; indeed, this assumption is incorporated into the normalization approach of Leblanc et al. (2010).

The initial loading stiffness is the obvious choice for estimating

k_{MAX}

, but there is significant variability in this value, probably due to bedding-in effects associated with the wished-in-place installation method. Maximum tangential stiffness values obtained from the first unloading portion (available for all cyclic tests) are more consistent and are therefore used here, although they may be affected by densification during initial (albeit small-amplitude) loading.

Fig. 6 presents the variation of

k_{MAX}

with stress level. Here,

k_{MAX}

is obtained by manual fitting to all relevant tests, and the range and mean values of

k_{MAX}

are indicated. A power function fits the variation of mean

k_{MAX}

with stress level well, as shown by the dotted line in Fig. 6. However, the exponent obtained from least-squares fitting is

η = 0.31

, somewhat lower than the

η = 0.5

for

G_{MAX}

that is implicit in the Leblanc et al. (2010) normalization. The cause of this discrepancy is unclear, but might be explained by experimental variability, the impact of stiffening during the initial loading portion, or three-dimensional effects when considering the monopile system. Using

η = 0.31

in place of

η = 0.5

in Normalization 3 produces a poorer result.

Cyclic Definitions

Loading

Constant-amplitude cyclic loading can be characterized by two parameters (Leblanc et al. 2010) as follows:

ζ_{b} = \frac{H_{MAX}}{H_{R}}

(1)

ζ_{c} = \frac{H_{MIN}}{H_{MAX}}

(2)

where

H_{MAX}

and

H_{MIN}

are the maximum and minimum loads applied in each cycle, respectively;

ζ_{c}

characterizes the load symmetry (

ζ_{c} = 0

for one-way loading,

ζ_{c} = - 1

for symmetric two-way loading, and

ζ_{c} = 1

for constant loading); and

ζ_{b}

and

ζ_{c}

together describe the load amplitude, relative to an arbitrary reference load

H_{R}

, where

H_{R}

is defined at a reference pile displacement value

u_{R} = 17.8 mm

at the load application point. This reference displacement approximately corresponds to 2° pile rotation (

θ_{R} = 2 °

) and

0.1 D

pile displacement at the mudline, which may represent an ultimate failure criteria; similar reference values have been used in studies by Abadie et al. (2015) and Arshad and O’Kelly (2017). Table 5 summarizes the monopile’s reference load values at each

g

-level obtained from initial monotonic tests.

Table 5. Reference load (

H_{R}

) values for the model monopile at each

g

-level

$g - level$ , $n$	$H_{R}$ (N)
1	15.7
9	99.5
80	597

The average load amplitude

H_{A V} = (H_{MAX} + H_{MIN}) / 2

and cyclic load amplitude

H_{CYC} = (H_{MAX} - H_{MIN}) / 2

are also useful parameters that can be used to describe the cyclic response.

Cyclic Response

The monopile’s cyclic response in dry sand is characterized in terms of permanent accumulation of displacement (ratcheting), evolution of secant stiffness, and evolution of energy dissipation per cycle.

Ratcheting is interpreted in terms of accumulated mean displacement at the load application point (

Δ u_{M}

), following the approach of Richards et al. (2019), but with displacement instead of rotation as the strain variable.

Secant stiffness

k_{s}

is defined at the center of the cycle (shown in Fig. 7 for Cycle 1) to minimize conflation of stiffness change with ratcheting. This stiffness is the inverse of the average of the loading and unloading flexibilities and may be expressed in terms of the secant loading stiffness

k_{s l}

and secant unloading stiffness

k_{s u}

, (each indicated in Fig. 7) as follows:

k_{s} = \frac{1}{\frac{1}{2} (\frac{1}{k_{s l}} + \frac{1}{k_{s u}})} = 2 (\frac{k_{s l} k_{s u}}{k_{s l} + k_{s u}})

(3)

Fig. 7. Cycle, stiffness, and energy loss factor definitions.

The energy dissipation is interpreted in terms of an energy loss factor

η

proportional to the ratio of hysteretic energy loss

Ω

to maximum stored energy per cycle

U_{MAX}

(Inman 2014) as follows:

η = \frac{Ω}{2 π U_{MAX}}

(4)

where the maximum stored energy per cycle can be related to the secant stiffness with:

U_{MAX} = \frac{1}{8 k_{s}} {(H_{MAX} - H_{MIN})}^{2}

(5)

For nonclosing hysteresis loops (due to ratcheting), the hysteretic energy loss across a cycle

i

is defined as follows:

Ω_{i} = \int_{r_{i}}^{m_{i}} H (u) d u + \int_{m_{i}}^{r_{i + 1}} H (u) d u

(6)

where the reversal (

r

) and maximum (

m

) points are indicated in Fig. 7. The energy loss factor becomes:

η = \frac{4 k_{s}}{π {(H_{MAX} - H_{MIN})}^{2}} (\int_{r_{i}}^{m_{i}} H (u) d u + \int_{m_{i}}^{r_{i + 1}} H (u) d u)

(7)

Unidirectional Cyclic Response

Test Program

Table 6 summarizes the unidirectional cyclic tests conducted as part of this campaign. One-way (

ζ_{c} = 0

) and two-way (

ζ_{c} = - 1

) tests were conducted, and the majority of tests involved 1,000 initial loading cycles followed by a reload to

0.8 H_{R}

to explore the postcyclic reloading response. The tests are characterized in terms of nominal and applied values of

ζ_{b}

and

ζ_{c}

. The nominal values are those demanded of the control system, and the applied loads are obtained from the in-line load measurements; the discrepancy is small and reduces with increasing load level as the relative size of temperature effects and the load cell signal-to-noise ratio reduces.

Table 6. Summary of unidirectional cyclic tests conducted

Test	$g - level$ , $n$	$ζ_{b}$		$ζ_{c}$		No. of cycles, $N$	Reload
Test	$g - level$ , $n$	Nominal	Applied	Nominal	Applied	No. of cycles, $N$	Reload
U.1.A	1	0.4	0.38–0.40	0	$- 0.03 - 0.02$	1000	Yes
U.1.B	1	0.3	0.28–0.30	0	$- 0.06 - 0.05$	770^a	Yes
U.1.C	1	0.2	0.18–0.17	0	$- 0.06 - 0.06$	1,000	Yes
U.1.D	1	0.4	0.40–0.42	$- 1$	$(- 0.97) - (- 0.89)$	500^b	No
U.9.A	9	0.4	0.39–0.40	0	$- 0.004 - 0.01$	1,000	Yes
U.9.B	9	0.3	0.29–0.30	0	$- 0.01 - 0.01$	1,000	Yes
U.9.C	9	0.2	0.19–0.20	0	$- 0.02 - 0.01$	1,000	Yes
U.9.D	9	0.4	0.39–0.40	$- 1$	$(- 1.02) - (- 0.99)$	1,000	Yes
U.80.A	80	0.4	0.39–0.40	0	$- 0.006 - 0.02$	1,000	Yes
U.80.C	80	0.2	0.20–0.20	0	$- 0.002 - 0.004$	1,000	Yes
U.80.D	80	0.4	0.40–0.40	$- 1$	$(- 1.01) - (- 0.99)$	1,000	Partial

a

Final 230 cycles of Test U.1.B excluded as they appear to be anomalous, probably due to temperature effects, which were significant at

1 g

.

b

Only 500 cycles completed for Test U.1.D due to problems with load control.

Load-Displacement Response

Fig. 8 presents the load-displacement (H-u) responses for the tests summarized in Table 6, alongside the corresponding monotonic responses (

M . n

). The full responses are shown for one-way tests

U . n . A

,

U . n . B

, and

U . n . C

, but only the first five cycles of two-way tests

U . n . D

are shown to highlight the loop shape. The responses are normalized by the load and displacement reference values (

H_{R}

and

u_{R}

) to present data from all three stress levels on comparable axes. The results show good repeatability at each stress level and qualitatively similar behavior across the three stress levels, although it is clear that the responses become more linear with increasing stress level. This variation in linearity is linked to stress dependent stiffness and dilatant behavior, as previously discussed, as well as the relative load amplitude. To better match the linearity of the cyclic responses across the stress levels, the reference pile displacement

u_{R}

—which determines

H_{R}

and therefore the cyclic amplitude

ζ_{b}

—may be adjusted with stress level, perhaps employing the dimensionless framework of Leblanc et al. (2010). However, the variation in backbone linearity is a stress-level effect, and it is not clear whether it is appropriate to eliminate this effect by modifying

u_{R}

.

Fig. 8. Unidirectional cyclic loading load-displacement responses: (a) one-way; and (b) two-way (first five cycles).

Fig. 8(b) also reveals evidence of gapping-like behavior in tests U.

n

.D at

1 g

and to a lesser extent at

9 g

. The inflexion in the load-displacement response around zero load is indicative of a gapping-type response because the tangent stiffness would reduce significantly when a pile traverses a gap. Similar load-displacement responses were recorded during field testing in both sand and clay as part of the pile soil analysis (PISA) project, where gapping was also observed on site (Beuckelaers 2017). Gapping was not observed during these tests, but the setup was not designed to record gap formation. A gap of only a fraction of a millimeter would be sufficient to account for the observed behavior at

1 g

.

For these tests, the gapping-like behavior reduces with increasing stress level. Some cohesion is necessary for gap formation, and so it is not generally expected in dry sand. However, the tendency for sand particles to move into a gap will reduce with reducing stress levels. At very low stress levels, electrostatic effects or ambient moisture may then provide enough cohesion for gap formation. The wished-in-place installation will also lead to very low horizontal stresses adjacent to the pile, increasing the likelihood of gap formation.

Ratcheting Response

Fig. 9(a) presents evolution of normalized accumulated displacement (

Δ u_{M} / u_{R}

) with cycle number (

N

) for the one-way unidirectional cyclic tests; a power law [Eq. (8)] is also fitted to each test response and shown as a dashed line. Various studies have shown ratcheting to evolve approximately as a power law with cycle number

N

(Leblanc et al. 2010; Klinkvort 2012; Abadie 2015; Albiker et al. 2017; Truong et al. 2019), and a power law is also suitable here at all stress levels:

Δ u_{M} / u_{R} = A N^{α}

(8)

Fig. 9. Unidirectional ratcheting response: (a) accumulation of displacement with cycle number $N$ ; (b) variation of ratcheting coefficient $A$ with $ζ_{b}$ ; and (c) variation of ratcheting exponent $α$ with stress level.

The power-law coefficient

A

has been found to vary with

ζ_{b}

and

ζ_{c}

(Leblanc et al. 2010; Nicolai and Ibsen 2014), and the power-law exponent

α

has often been reported as a constant within individual studies (Leblanc et al. 2010; Nicolai and Ibsen 2014; Albiker et al. 2017). However, Truong et al. (2019) suggested that

α

varies with

ζ_{c}

and

D_{R}

. This paper investigates only a single value of

ζ_{c}

and

D_{R}

.

No dependence of

A

on stress level is observed, although Fig. 9(b) shows how

A

varies with

ζ_{b}

as a power law, in line with Abadie (2015) as follows:

A = ζ_{b}^{0.34}

(9)

Conversely, no dependence of

α

on

ζ_{b}

is observed (in agreement with previous studies), but there is a trend for decreasing

α

with increasing stress level, as shown in Fig. 9(c). To facilitate prediction of behavior at other stress levels, the ratcheting exponent

α

is presented in Fig. 9(c) against normalized reference stress level

σ_{REF}^{'} / p_{a}

rather than

g

-level.

A logarithmic trend line is fitted to the data in Fig. 9(c) to quantify the variation of

α

with stress level and is shown as a dashed line. The line is defined by:

α = 0.127 - 0.022 \ln (\frac{σ_{REF}^{'}}{p_{a}})

(10)

This trend line suggests that for a full-size monopile in dense saturated sand (

D = 8 m

,

γ^{'} = 10 kN / m^{3}

, and

σ_{REF}^{'} / p_{a} = 2.2

), the ratcheting exponent may be as low as 0.11, approximately half of the value in the

1 g

model tests.

The value of

α

appears to vary with the chosen strain variable, and it is therefore difficult to compare

α

values from independent tests campaigns to build confidence in this conclusion on stress dependency; for example, Albiker et al. (2017) found

α = 0.23

with pile rotation as the strain variable and

α = 0.135

with pile displacement (some distance above mudline) as the strain variable. However, the results presented here appear to be consistent with the increase in normalized pile displacement for

1 g

test compared with centrifuge tests reported by Rudolph et al. (2014a, b).

Secant Stiffness Response

Fig. 10(a) presents evolution of the monopile’s secant stiffness

k_{s}

with cycle number for all unidirectional cyclic tests. Stiffness is normalized by the maximum stiffness

k_{MAX}

at each

g - level

, obtained from the first-cycle unloading portion. An increase in secant stiffness with cycle number is observed for all tests, with

k_{s} / k_{MAX}

plateauing at high cycle number in all tests except U.1.D.

Fig. 10. Unidirectional secant stiffness response: (a) evolution of secant stiffness with cycle number $N$ ; (b) variation of initial secant stiffness with stress level; and (c) variation of secant stiffness exponent $β$ with stress level.

The initial normalized secant stiffness [

k_{s (N = 1)} / k_{MAX}

] is plotted in Fig. 10(b). There is no strong dependence of general initial normalized secant stiffness values on stress level; however, the spread of initial stiffness values decreases with increasing stress level, consistent with the increasing linearity of the responses. At each

g

-level, the initial stiffness values generally decrease with cyclic amplitude

H_{CYC}

, as expected, given the nonlinear load-displacement response.

Evolution of secant stiffness with cycle number has previously been described by logarithmic functions (Klinkvort and Hededal 2013; Leblanc et al. 2010; Abadie 2015). Although a logarithmic function captures these data reasonably well, a power law [Eq. (11)] fits equally well, and is preferred here for consistency of interpretation with ratcheting. However, neither function captures the response particularly well for

N < 10

. Power-law fits are shown by a dashed line in Fig. 10(a) following:

k_{s} / k_{MAX, n} = B N^{β}

(11)

The impact of stress level on the evolution of stiffness is assessed by plotting the power-law exponent (

β

) against stress level in Fig. 10(c). The exponents for tests U.1.D (

β = 0.304

) and U.9.D (

β = 0.110

) lie off this plot, being considerably greater than the other exponents. These high values of

β

may be linked to the gapping-like behavior observed for these tests. If a gap is opening, the possibility of grain migration, and therefore densification close to the pile, is increased.

A logarithmic trend line is fitted to the variation of

β

with stress level, neglecting the two-way tests. The dashed trend line in Fig. 10(c) has the following equation:

β = 0.017 - 0.0076 \ln (\frac{σ_{REF}^{'}}{p_{a}})

(12)

This trend line suggests that for a full-size monopile, the one-way stiffness exponent

β

may be as low as 0.011, or 25% of the stiffness exponent value in the

1 g

model tests.

The power-law function for

k_{s}

[Eq. (11)] is not bounded as

N

approaches infinity. However, given the small magnitude of

β

at the stress levels of interest, reasonable values for secant stiffness can be expected for all practical cycle numbers.

Energy Loss Factor Response

Fig. 11(a) shows the evolution of unidirectional energy loss factor with cycle number. Abadie (2015) also explored the evolution of an energy loss factor with cycle number and found that a power law adequately captured the evolution. A power law [Eq. (13)] is also found to provide a good fit to the energy loss factor evolution here at all stress levels, and the resulting fits are shown by a dashed line in Fig. 11(a):

η = C N^{γ}

(13)

Fig. 11. Unidirectional energy loss factor: (a) evolution of energy loss factor with cycle number $N$ ; (b) variation of initial energy loss factor with stress level; and (c) variation of energy loss factor exponent $γ$ with stress level.

For ratcheting and stiffness evolution, the variation of the energy loss factor exponent

γ

with stress level is plotted in Fig. 11(c). However, no dependence of

γ

on stress level is observed. Instead, the initial energy loss factor

η_{N = 1}

is found to decrease with increasing stress level, as shown in Fig. 11(b). Similar behavior would be observed by plotting coefficient

C

against stress level. The anomalously low value of initial energy loss factor for Test U.1.D is linked to the significant gapping-like behavior observed, which reduces the hysteretic energy loss

Ω

compared with no gapping.

A logarithmic function is fitted to the variation of initial energy loss factor

η_{N = 1}

with stress level. The following equation defines this trend line:

η_{N = 1} = 0.228 - 0.134 \ln (\frac{σ_{REF}^{'}}{p_{a}})

(14)

This trend line suggests an initial energy loss factor of around 0.12 for an equivalent full-size monopile, which is 16% of the value in the

1 g

model tests.

As for secant stiffness, the power-law function for

η

[Eq. (13)] is not bounded as

N

approaches infinity. However, given the small magnitude of

γ

at the stress levels of interest, reasonable (nonnegligible) values of energy loss factor are expected for all practical cycle numbers.

Reloading Response

Postcyclic reloading was performed after 1,000 loading cycles for the majority of the unidirectional cyclic tests. Fig. 8(a) shows how the reloading response tends to rejoin the backbone curve following one-way cyclic loading at all stress levels. It is understood that ratcheting and stiffening mechanisms act in competition, often resulting in no significant net effect on the displacement response at large loads (Abadie et al. 2019). The reloading response is also presented in Fig. 12 with the displacement rezeroed (from displacement at start of reload

u_{r 0}

). This plot shows how the reloading response is significantly stiffer than the initial loading response for

H / H_{R} < ζ_{b}

, consistent with the observed increase in secant stiffness with cycling. In general, when compared with the corresponding backbone curves, the reloading responses are qualitatively similar at all stress levels.

Fig. 12. Rezeroed reloading response following unidirectional cyclic loading.

Discussion

Qualitatively, the unidirectional cyclic response is similar across the three stress levels investigated, with similar reloading responses and with power-law expressions approximately capturing the evolution of ratcheting, stiffness, and energy loss factor at all stress levels. Quantitatively, however, the results reveal some important stress-level effects.

The decrease of initial energy loss factor

η_{N = 1}

with stress level can be directly linked to the linearity of the load-displacement response. The response linearity increases with stress level for a given normalized load amplitude

ζ_{b}

and is linked to stress dependent stiffness and dilatant behavior. However, the logarithmic decrease in ratcheting and stiffness exponent with stress level [Figs. 9(c) and 10(c)] appears to be independent of load amplitude and therefore of linearity of the load-displacement response.

Cuéllar et al. (2012) and Nicolai (2017) explored the mechanism driving ratcheting and stiffening behavior for cyclically loaded monopiles in sand. Cuéllar et al. (2012) used colored sand grains at

1 g

, and Nicolai (2017) used particle image velocimetry in the centrifuge. Both authors observed local sand densification and convective particle movements during cyclic loading, and Cuéllar et al. (2012) identified three-dimensional (3D) convective cells. Cui and Bhattacharya (2016) conducted distinct element method (DEM) simulations to explore the same problem and also observed convective particle movement and densification. Cui and Bhattacharya (2016) also suggested that the convective region is the shape of an inverted cone [also observed by Cuéllar (2011)] because increasing stress level with depth imposes increasing constraint on particle movement.

The increase in the monopile’s secant stiffness is probably caused by a local densification mechanism, and the ratcheting behavior is probably caused by both a three-dimensional convective mechanism and any asymmetry in local densification. Given that both local densification and convective mechanisms require particle rearrangement, it is explicable that these mechanisms, and associated ratcheting and stiffening behavior, will be affected by increasing stress level, where—as highlighted by Cui and Bhattacharya (2016)—there is increasing constraint on particle rearrangement.

Multidirectional Cyclic Response

Test Program and Loading Definition

Table 7 summarizes the multidirectional cyclic tests performed as part of this campaign. Two multidirectional tests were performed at each

g

-level: a T-shaped test and an L-shaped test. These tests explore the impact of cyclic load and load bias direction and are intended to be compared with the one-way unidirectional tests at the same cyclic load amplitude

H_{CYC}

and average load amplitude

H_{A V}

.

Table 7. Summary of multidirectional cyclic tests conducted

Test	$g$ -level, $n$	$I$ -direction				II-direction				No. of cycles, $N$
		$ζ_{b}$		$ζ_{c}$		$ζ_{b}$		$ζ_{c}$
		Nominal	Applied	Nominal	Applied	Nominal	Applied	Nominal	Applied
M.1.T	1	0.35	0.37–0.39	1	0.80–1.05	0.2	0.19–0.22	$- 1$	$(- 1.20) - (- 0.80)$	250^a
M.1.L	1	0.35	0.36–0.39	1	0.98–1.08	0.4	0.37–0.38	0	$(- 0.07) - (0.00)$	153^a
M.9.T	9	0.2	0.20–0.23	1	0.95–0.99	0.2	0.19–0.20	$- 1$	$(- 1.03) - (- 0.99)$	1,000
M.9.L	9	0.2	0.21–0.24	1	0.93–0.97	0.4	0.39–0.38	0	$(- 0.06) - (0.00)$	1,000
M.80.T	80	0.2	0.20–0.22	1	0.95–1.00	0.2	0.20–0.20	$- 1$	$(- 1.00) - (- 0.95)$	1,000
M.80.L	80	0.2	0.20–0.22	1	0.95–1.00	0.4	0.39–0.39	0	$(- 0.02) - (0.00)$	1,000

a

Fewer than 1,000 cycles completed for these tests due to load control problem.

Fig. 13 shows the loading pattern for each test type: the selected unidirectional tests involve one-way cycling (

ζ_{c} = 0

), T-shaped tests involve symmetric cycling (

ζ_{c} = - 1

) perpendicular to a constant-load bias, and L-shaped tests involve one-way cycling (

ζ_{c} = 0

) perpendicular to a constant-load bias. Unidirectional, T-shaped, and L-shaped tests therefore have average load biases in the

I I

-direction,

I

-direction, and at 45° to the axes respectively, following the axes in Fig. 13. The T- and L-shaped tests may represent misaligned wave and wind loading on an OWT, where cyclic loading approximates wave loading and the load bias approximates the more slowly varying wind loading.

Fig. 13. Schematic demonstrating multidirectional loading.

Table 7 characterizes the tests in terms of nominal and applied values of

ζ_{b}

and

ζ_{c}

. The nominal values are those demanded of the control system during testing without considering load adjustment due to pile movement. The applied values are resolved from the load cell measurements posttest, accounting for the current pile position and load line angle. The multidirectional tests at

9 g

and

80 g

correspond to unidirectional tests U.

n

.A conducted at the same cyclic load amplitude

H_{CYC}

and average load amplitude

H_{A V}

. The multidirectional

1 g

tests have a nominal load bias in the

I

-direction 75% larger than U.1.A, and comparisons at

1 g

should therefore be made with caution.

Displacement and Ratcheting Response

Fig. 14 presents the displacement response for the multidirectional tests summarized in Table 7 alongside the corresponding unidirectional tests. The displacements at cycles 1, 10, 100, (and 1,000) are marked. Fig. 14 shows how the monopile moves broadly in the direction of the load bias at all stress levels, though there is some deviation in M.9.T and M.80.T. Fig. 14 also highlights the increase in amplitude of cyclic displacement with stress level, which is in line with the changing linearity of the load-displacement response with stress level.

Fig. 14. Displacement response for multidirectional tests. Markers indicate displacement at Cycles 1, 10, 100, (and 1,000).

Fig. 15(a) presents the accumulated displacement response for the multidirectional tests alongside the corresponding unidirectional tests. Multidirectional behavior is presented in terms of orthogonal

I

- and II-direction components. For the T-shaped tests, significant ratcheting only occurs and is only reported in the

I

-direction. For the L-shaped tests ratcheting is reported in both the

I

- and II-directions. A power law [Eq. (8)] is fitted to the evolution of

Δ u_{M} / u_{R}

with cycle number and shown as a dashed line in Fig. 15(a). The power law generally captures the evolution of ratcheting well for

N > 10

, but tends to overpredict ratcheting for

N < 10

for the T-shaped test. The response of test M.80.L(I), particularly for

N > 100

, may be anomalous.

Fig. 15. Multidirectional ratcheting response: (a) accumulation of displacement with cycle number $N$ ; (b) variation of ratcheting coefficient $A$ with multidirectional loading type; and (c) variation of ratcheting exponent with stress level.

Fig. 15(b) presents the variation of ratcheting coefficient

A

with test type and stress level. No clear dependence of

A

on either test type or stress level is observed. The ratcheting exponent

α

is plotted against stress level in Fig. 15(c), accompanied by the dashed trend line obtained for the unidirectional tests [Eq. (10)]. This trend line also fits the multidirectional data well. In general, Fig. 15 shows no clear dependency of ratcheting behavior on test type (at least for

N > 10

).

The greater scatter in Figs. 15(b and c) for the

1 g

tests is explained by the inconsistency in applied load bias.